Chapter 9, Problem 8MCQ

(a)

To determine

Find the height of the ball after one second.

Answer to Problem 8MCQ

After 1 second the ball will reach 76 feet.

Explanation of Solution

Given:

A soccer ball is kicked from ground level with a initial upward velocity of 90 feet per second. The equation $h=-16t^2+90t$ gives the height h of the ball after t seconds.

Concept Used:

Find the height of the ball after one second. This means we need to substitute t = 1 in the equation $h=-16t^2+90t$

Calculation:

  $\begin{array}{l}h=-16t^2+90t\\ $ h=-16{(1)}^2+90·1=-16+90=76$\end{array}$

After 1 second the ball will reach 76 feet.

Thus, after 1 second the ball will reach 76 feet.

(b)

To determine

Find the time in seconds it will take for the ball to reach its maximum height.

Answer to Problem 8MCQ

Time taken to reach the maximum height is 2.8 seconds.

Explanation of Solution

Given:

A soccer ball is kicked from ground level with a initial upward velocity of 90 feet per second. The equation $h=-16t^2+90t$ gives the height h of the ball after t seconds.

Concept Used:

The graph of the equation the equation $h=-16t^2+90t$ is a downward parabola. For downward parabola vertex is maximum. x coordinate of the vertex of $h=-16t^2+90t$ is the time taken to reach the maximum height.

Calculation:

The equation $h=-16t^2+90t$

The x- coordinate of the parabola is $x=-\frac{b}{2a}$

Find the x − coordinate of the parabola equation $h=-16t^2+90t$

  $x=-\frac{b}{2a}=-\frac{90}{2·-16}=-\frac{90}{-32}=\frac{90}{32}=\frac{45}{16}=2.8125$

x − coordinate is the time taken to reach the maximum height.

Time taken to reach the maximum height is 2.8 seconds.

Thus, time taken to reach the maximum height is 2.8 seconds.

(c)

To determine

The time when the ball will reach zero feet and the meaning of these points represent in this situation.

Answer to Problem 8MCQ

The time 5.6 seconds when the ball will reach zero feet.

Explanation of Solution

Given:

A soccer ball is kicked from ground level with a initial upward velocity of 90 feet per second. The equation $h=-16t^2+90t$ gives the height h of the ball after t seconds.

Concept Used:

The time when the ball will reach zero feet means substitute h = 0

Calculation:

Rewrite the equation by substituting h = 0

  $\begin{array}{l}h=-16t^2+90t\\ 0=-16t^2+90t\\ -16t^2+90t=0\end{array}$

Solve for t :

  $\begin{array}{l}-16t^2+90t=0\\ -2t(8t-45)=0[\text{Factor}\text{it}]\\ \text{Use}\text{the}\text{xero}\text{product}\text{rule}\\ -2t=0\text{and}8t-45=0\\ -2\ne 0;t=0\text{and}8t-45=0\\ 8t-45=0\\ 8t=45[\text{add}\text{45}\text{to}\text{both}\text{sides}]\\ t=\frac{45}{8}=5.625\end{array}$

  $t=0\text{and}t=5.6$

Time $t=0$ second corresponds to when the soccer ball is hit from the ground, so $t\approx 5.6$ seconds corresponds to when it lands on the ground.

Thus, the time 5.6 seconds when the ball will reach zero feet.